Marginal Convergence with Independence Implies Joint Convergence Consider a a sequence of random vectors $\{X_n\}$, where each $X_n = \begin{bmatrix} X_{1, n} \\ \vdots \\ X_{m, n} \end{bmatrix}$. Define $X_0 = \begin{bmatrix} X_{1, 0} \\ \vdots \\ X_{m, 0} \end{bmatrix}$.
Now, if $X_{j, n}$ converges in distribution to $X_{j, 0}$ for $j = 1, \dots m$ (marginal convergence), and moreover the $X_{j, n}$ ($j = 1, \dots, m$) are independent from each other for all $n$, then I'm looking to show that we will also have $X_n$ converging to $X_0$ in distribution.
I've gone about this without any knowledge or results from measure theory (i.e. some basic analysis tools regarding convergence of sequences). In particular, for a fixed arbitrary $\epsilon > 0$, I'm trying to show that there exists some $N \in \mathbb{N}$ such that $\forall n > N$ we have $|X_n - X_0| < \epsilon$. But I'm just playing with norm inequalities without using the fact that the $X_{j, n}$ are independent from each other, so I've clearly gone wrong. Any pointers?
Note that there is a similar question here which deals with characteristic functions that I am unfamiliar with.
 A: The standard device for establishing convergence in distribution of $m$-dimensional random vectors $(X_n)$ to a limit $X_0$ is known as the Cramer-Wold theorem, which reduces convergence in $R^m$ to convergence of scalar-valued random variables:

Theorem: $X_n$ converges in distribution to $X_0$ if and only if $v^TX_n$ converges in distribution to $v^TX_0$ for every $v\in R^m$.

This is what justifies the characteristic-function argument you cited -- independence of the components of $X_n$ allows the characteristic function of $v^TX_n$ to be factored, which then easily establishes the Cramer-Wold condition. [If you are unfamiliar with characteristic functions, they are the complex-valued analog of the moment generating function $M_X(t):=E(\exp (tX))$, with the added benefit that the characteristic function $\phi_X(t):=E(\exp (itX))$ is defined for all $t\in R$ whereas the moment generating function may not exist for all $t$.]
An alternative to the Cramer-Wold approach is the following theorem, which is the vector analogue of the definition of convergence in distribution for random variables. It makes use of the $m$-dimensional distribution function, defined for any random vector $X:=(X_1,\ldots,X_m)$:
$$ F_X(x) := F_X(x_1,\ldots,x_m):=P( X_1\le x_1, \ldots, X_m\le x_m)$$

Theorem: $X_n$ converges in distribution to $X_0$ if and only if $F_{X_n}(x)\to F_{X_0}(x)$ for every $x\in R^m$ at which $F_{X_0}(x)$ is continuous.

This approach works well in your case where the vector components are independent, since the $m$-dimensional distribution function factors into its scalar components.
